The red graph (on three vertices) in figure REF  is a degeneration of both the blue graph as well as the green graph, both on four vertices. Note that in Morse theory, the change in topology going from the left pair of circles to the right circle is detected by the index of the saddle point. Such saddle points are known to characterize transitions in various contexts in physics. For instance, in dynamical systems with a few degrees of freedom, a diagram such as figure REF  can represent the change in the topology of the phase portrait of the system due to a saddle point in the energy of the system (a Morse function). This change represents a dynamical transition (or bifurcation) [1]}, [2]}. An example involves the stabilisation of the inverted pendulum [3]}. Analogously, in the Ginzburg-Landau-Wilson approach to critical phenomena [4]}, saddle points in the free energy of a system in which a macroscopic number of degrees of freedom are interacting with one another can signify a phase transition (i.e., a transformation of the phase of the system). Here, the free energy is written in terms of a coarse-grained order parameter determined purely by the symmetries and spatial dimensionality of the system. Another important example involves the case of Lifshitz transitions, i.e., phase transitions that involves changes in the Euler characteristic of the zero-temperature Fermi volume of a system of electrons [5]}. Using the fact that the electronic dispersion relation \(E_{\vec{k}}\)  (i.e., the energy-momentum relation) for a system of electrons is a Morse function, the Euler characteristic of the Fermi volume (i.e., the set of all electronic states that are occupied at zero temperature) can be obtained from the critical points of \(E_{\vec{k}}\)  by the application of Morse theory [6]}. This Euler characteristic can then be shown to change when the application of external parameters (e.g., pressure, chemical potential etc.) tunes extrema of \(E_{\vec{k}}\)  (e.g., saddles points, maxima and minima) through the Fermi energy \(E_{F}\)  (i.e., the greatest occupied energy state, and denoting the Fermi surface in the \(D+1\)  dimensional energy-momentum space of a \(D\) -dimensional system of electrons). These extrema are the singularities of the Morse function \(E_{\vec{k}}\) , and reflect the existence of van Hove singularities in the electronic density of states [6]}. While such Lifshitz transitions have typically been studied in systems of non-interacting electrons, a lot of interest in quantum condensed matter physics presently involves searching for similar Lifshitz transitions of the Fermi volume in strongly correlated electronic systems [8]}, [9]}.
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